GPS has better than 10m accuracy in another operating mode. Only the military is allowed to use that one, though. Then they don't want Galileo being 1m accurate? Probably same military reasons.
No, that's not true. The military has access to different codes and frequencies that aren't available for civilian use, which can be used to apply further corrections to the computed location. Selective Availability was not the only difference between civilian and military GPS.
As far as I understand, the remaining differences between military and civilian GPS don't matter much for accuracy. The military may still have the advantage of ionospheric correction, but civilians can use other augmentation systems like WAAS[1] with similar results. I think the military signals are mostly intended to protect against signal spoofing and jamming.
Those extra frequencies were what I was thinking about. If they're still closed, then military still cripples GPS. The points of military vs civilian for this discussion would still apply.